We present a novel approach for nonlinear, three dimensional deformation of a rod that allows in-plane cross-sectional deformation. The approach is based on the concept of multiplicative decomposition, i.e., the deformation of a rod’s cross section is performed in two steps: pure in-plane cross-sectional deformation followed by its rigid motion. This decomposition, in turn, allows straightforward extension of the special Cosserat theory of rods (having rigid cross section) to a new theory allowing in-plane cross-sectional deformation. We then derive a complete set of static equilibrium equations along with the boundary conditions necessary for analytical/numerical solution of the aforementioned deformation problem. A variational approach to solve the relevant boundary value problem is also presented. Later we use symmetry arguments to derive invariants of the objective strain measures for transversely isotropic rods, as well as for rods with inbuilt handedness (hemitropy) such as DNA and carbon nanotubes. The invariants derived put restrictions on the form of the strain energy density leading to a simplified form of quadratic strain energy density that exhibits some interesting physically relevant coupling between the different modes of deformation.
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January 2011
Research Papers
A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation
Ajeet Kumar,
Ajeet Kumar
Postdoctoral Research Associate
Department of Aerospace Engineering & Mechanics,
University of Minnesota
, Minneapolis, MN 55455
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Subrata Mukherjee
Subrata Mukherjee
Professor
Fellow ASME
Sibley School of Mechanical and Aerospace Engineering,
Cornell University
, Ithaca, NY 14853
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Ajeet Kumar
Postdoctoral Research Associate
Department of Aerospace Engineering & Mechanics,
University of Minnesota
, Minneapolis, MN 55455
Subrata Mukherjee
Professor
Fellow ASME
Sibley School of Mechanical and Aerospace Engineering,
Cornell University
, Ithaca, NY 14853J. Appl. Mech. Jan 2011, 78(1): 011010 (10 pages)
Published Online: October 13, 2010
Article history
Received:
November 25, 2009
Revised:
May 18, 2010
Posted:
June 9, 2010
Published:
October 13, 2010
Online:
October 13, 2010
Citation
Kumar, A., and Mukherjee, S. (October 13, 2010). "A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation." ASME. J. Appl. Mech. January 2011; 78(1): 011010. https://doi.org/10.1115/1.4001939
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